Computing comprehensive Gröbner systems and comprehensive Gröbner bases simultaneously

Abstract

In Kapur et al (ISSAC, 2010), a new method for computing a comprehensive Grobner system of a parameterized polynomial system was proposed and its efficiency over other known methods was effectively demonstrated. Based on those insights, a new approach is proposed for computing a comprehensive Grobner basis of a parameterized polynomial system. The key new idea is not to simplify a polynomial under various specialization of its parameters, but rather keep track in the polynomial, of the power products whose coefficients vanish; this is achieved by partitioning the polynomial into two parts-nonzero part and zero part for the specialization under consideration. During the computation of a comprehensive Grobner system, for a particular branch corresponding to a specialization of parameter values, nonzero parts of the polynomials dictate the computation, i.e., computing S-polynomials as well as for simplifying a polynomial with respect to other polynomials; but the manipulations on the whole polynomials (including their zero parts) are also performed. Grobner basis computations on such pairs of polynomials can also be viewed as Grobner basis computations on a module. Once a comprehensive Grobner system is generated, both nonzero and zero parts of the polynomials are collected from every branch and the result is a faithful comprehensive Grobner basis, to mean that every polynomial in a comprehensive Grobner basis belongs to the ideal of the original parameterized polynomial system. This technique should be applicable to other algorithms for computing a comprehensive Grobner system as well, thus producing both a comprehensive Grobner system as well as a faithful comprehensive Grobner basis of a parameterized polynomial system simultaneously. The approach is exhibited by adapting the recently proposed method for computing a comprehensive Grobner system in (ISSAC, 2010) for computing a comprehensive Grobner basis. The timings on a collection of examples demonstrate that this new algorithm for computing comprehensive Grobner bases has better performance than other existing algorithms.